All Topics In Determinants

A square matrix’s determinant is calculated using different methods. It is the algebraic sum of the entities’ products in the matrix. Sometimes, the values are also calculated after row and column reduction. This helps to simplify the determination process and make it time-efficient. There are various properties of determinants, which are used to make their solving easier. Some determinants are also solved using the standard results that we already have. It is usually used to solve the equations using Cramer’s rule, wherein the matrix elements are the coefficients of the equation.

How to solve a determinant

Let’s say we have a matrix A given by

The most simple way of calculating the determinant would be

det(A)=| 3(7×2 – 7×8) -1(4×2 – 7×6) + 3(4×8 – 7×6) | = 122

In general form, a matrix would be like

The formula for the determinant would be

Det(A)= a11(a22xa33 – a32xa23) -a12(a21xa33 – a31xa23) + a13(a31xa22 – a32xa21)

Determinant of any matrix B can be presented as Det(B) or |B|.

Properties of determinants

The below properties are beneficial for determining the determinants.

• If In  is an identity matrix with order nxn, the value of det(I)= 1.

An identity matrix is a matrix with all the diagonal elements equal to 1, and the rest are 0.

• Det(AT)=Det(A)

The value of the determinant of the transpose of A is equal to the original value of determinant A.

• |MN|=|M||N|

The determinant value of two multiplied determinants is equal to the particular product of their determinants.

• |kA|=kn |A|, where n is the order of the square matrix A.

This property states that if a constant is multiplied by a matrix, then its determinant’s value is the product of the original value of the determinant and the constant is raised to the power of the order of the matrix.

• |-A|= (-1)n|A|, where n is the order of the matrix.
• The value of the determinant of a matrix is zero if the value of all its elements is zero.
• The determinant of a triangular matrix is the product of all its diagonal elements.

The matrix could be an upper or lower triangular matrix. An upper triangular matrix is where all the elements below the diagonal are zero.

• If B-1 is the inverse of the matrix B, then det(B-1)= 1/|B|.

Some more properties of determinants

• All-zero property

If all the entities in a matrix are zero, then the value of the determinant is also zero.

• Reflection Property

This property states that if the rows are changed to columns, and the columns are changed to rows, it does not change the value of the determinant. 

• Switching Property

If any two rows or columns are reversed, the sign of the final determinant value changes.

• Proportionality Property

This property states that if two or more or two or more columns are proportional, then the value of the determinant becomes zero.

• Triangle Property

The value of determinant in both the cases would be a11a22a33

• Invariance Property

If a row or column in a matrix is added with an equimultiple of another row or column, the value of the determinant does not change. 

Ri→Ri+bRj

• Sum Property

This property states that the sum of the product of the entities of any row or column with the cofactors of the corresponding elements of any other row or column is zero in a determinant. 

• Scalar Multiple Property

This property states that if all the entities of the rows or columns are multiplied by any constant, then the determinant would also be multiplied by the same constant.

• Factor Property

If the value of all the entities as k makes the value of determinant =0,

   i.e. x=k, x-k=0 is a factor of the determinant,

All these properties of the determinant are derived using the main formulas but can be used straight away in questions.

There are more properties of determinants that are used based on the type of matrix it is.

These properties help in the determination of the determinant.

Conclusion

The topic of determinants is very vast with many properties and short methods which make its determination easier and efficient, thus reducing the amount of time required to solve them. To learn about determinants, one must have enough knowledge about matrices. The types of matrices are their properties, and the properties of determinants are interrelated. The purpose of determinants is to make the solving process of nonhomogeneous equations easier by Cramer’s rule. The coefficients of the equation make the elements of the matrix. Questions based on determinants are matrices are asked in various examinations, which require the basic knowledge about the properties of determinants discussed above.